Theorem btwnlng1

Description: Betweenness implies colinearity. (Contributed by Thierry Arnoux, 28-Mar-2019)

	Ref	Expression
Hypotheses	btwnlng1.p	$$⊢P={\mathrm{Base}}_G$$
	btwnlng1.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
	btwnlng1.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
	btwnlng1.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
	btwnlng1.x	$$⊢\phi \to X\in P$$
	btwnlng1.y	$$⊢\phi \to Y\in P$$
	btwnlng1.z	$$⊢\phi \to Z\in P$$
	btwnlng1.d	$$⊢\phi \to X\ne Y$$
	btwnlng1.1	$$⊢\phi \to Z\in \left(XIY\right)$$
Assertion	btwnlng1	$$⊢\phi \to Z\in \left(XLY\right)$$

Proof

Step	Hyp	Ref	Expression
1		btwnlng1.p	$$⊢P={\mathrm{Base}}_G$$
2		btwnlng1.i	$$⊢I=\mathrm{Itv}\left(G\right)$$
3		btwnlng1.l	$$⊢L={Line}_{𝒢}\left(G\right)$$
4		btwnlng1.g	$$⊢\phi \to G\in {𝒢}_{\mathrm{Tarski}}$$
5		btwnlng1.x	$$⊢\phi \to X\in P$$
6		btwnlng1.y	$$⊢\phi \to Y\in P$$
7		btwnlng1.z	$$⊢\phi \to Z\in P$$
8		btwnlng1.d	$$⊢\phi \to X\ne Y$$
9		btwnlng1.1	$$⊢\phi \to Z\in \left(XIY\right)$$
10	9	3mix1d	$$⊢\phi \to \left(Z\in \left(XIY\right)\vee X\in \left(ZIY\right)\vee Y\in \left(XIZ\right)\right)$$
11	1 3 2 4 5 6 8 7	tgellng	$$⊢\phi \to \left(Z\in \left(XLY\right)\leftrightarrow \left(Z\in \left(XIY\right)\vee X\in \left(ZIY\right)\vee Y\in \left(XIZ\right)\right)\right)$$
12	10 11	mpbird	$$⊢\phi \to Z\in \left(XLY\right)$$